Internal Elastic Fields and The Dislocation Density Tensor in Deformed Internal Elastic Fields and The Dislocation Density Tensor in Deformed FCC Crystals: Computational Modeling and Experimental MeasurementsFCC Crystals: Computational Modeling and Experimental Measurements

MotivationMotivation

Boundary Value Problem of DislocationsBoundary Value Problem of Dislocations

Statistical ModelingStatistical Modeling

Experimental MeasurementExperimental Measurement

Preliminary ResultsPreliminary Results

DiscussionDiscussion

Jie Deng, Mamdouh Mohamed, Anter El-Azab (FSU) and Ben Larson (ORNL)

A volume of deformed crystal showing mesoscale deformation features

Resolved region showing complexdislocation structure

Internal stress of dislocation system exhibits statistical features

• Dislocation stress in a bounded crystal volume consists of two contributions:

• Infinite-domain solution is obtained from non-singular analytic formula:

• Image stress is the solution of a boundary value problem of dislocations:

• Internal elastic strain and lattice rotations by dislocations are part of the elastic solution.

• The dislocation density tensor is determined by lattice curvature and elastic strain gradients in the infinitesimal distortion regime:

( ) ( ) ( )rrr imgσσσ += ∞

( ) αββαα β πµ

πµσ rdRebrdRebr appiimmCappiimmC

ns ′∂ ′∂ ′∂ ′−′∂ ′∂ ′∂ ′−= ∫∫∞

88,

( ) ( ) kappiaiimkmCrdRReb ′∂ ′∂ ′∂ ′−∂ ′∂ ′∂ ′

−− ∫ α ββα δ

νπµ14

( ) ( ) 0=⋅∇+⋅∇ ∞ rrimg σσ with boundary conditions ( ) ( ) ( ) ( )rnrrnrimg ∞−= σσ

ljkiklkkijjiij e εκδκα ∂−−=

• Mesoscale plastic deformation of crystals is governed by motion and interaction of large dislocation systems.

• The evolution of dislocation systems is connected to their internal elastic fields, which exhibit a statistical nature due to dislocation density fluctuations.

• 3D X-ray microscopy techniques now have the capability to measure internal elastic fields and dislocation density tensors with sub-micrometer resolution, making possible direct comparison with dislocation dynamics deformation simulations.

• Understanding the statistics of internal elastic fields and the dislocation density tensor helps complete the mesoscale crystal plasticity theory.

• Here we present initial investigation of these statistics using both computer simulations and experimental measurements.

• Probability distribution for internal stress fields of dislocations in deformed Cu

• Pair correlation of internal stress fields of dislocations in deformed Cu

• Probability distribution of the dislocation density tensor in deformed Cu

• Curvature contribution to the dislocation density tensor in deformed Cu

• Submicron (0.5 µm) 3D X-Ray Microscopy Measurement of Rotation Distortions in deformed Cu

• The statistics of internal elastic fields and dislocation density tensor has been modeled by generalized n-th order probability density function of dislocation density

• First order probability density function of internal elastic fields:

• Pair correlation function of internal elastic fields:

• First order probability density function of dislocation density tensor:

• Higher order probability density functions and correlation functions can be formulated by following the same strategy.

( ) ( )nnss rrf n θθ ,,,, 11

,,1 ⋅⋅⋅⋅⋅⋅

( ) ( ) ( ) ( )[ ] nnijnnss

ssij ddrddrrrrfrp n

n

θθββδθθβ ⋅⋅⋅−⋅⋅⋅= ⋅⋅⋅

Ω⋅⋅⋅

∫∑ 1111,,

,,,,,,, 1

1

°° ( ) ( ) Vrpp ijVij /,°° ββ ∫=

( ) ( ) ( ) ( ) ( )( )''', rrrrrrC klijklijijkl ββββ= ( ) ( ) ( ) ( ) ( )rrrrrrrC klijklijijkl ∆+∆+=∆ 22 ββββ

( ) ( ) ( ) ( )[ ] nnijnnss

ssij ddrddrrrrfrp n

n

θθααδθθα ⋅⋅⋅−⋅⋅⋅= ⋅⋅⋅

Ω⋅⋅⋅

∫∑ 1111,,

,,,,,,, 1

1

°° ( ) ( ) Vrpp ijVij /,°° αα ∫=

Differential-Aperture 3D X-Ray Microscopy*

CCD

K-B FocusingMirrors

[~0.5 µm Beam] RemovableMonochromator

Differential-Aperture (Wide-Angle) Profiler

Bulk Sample

Synchrotron PolychromaticX-Ray Beam

[Larson et al., Nature 415, 887 (2002)]

• White Beams Generate Full Laue Diffraction Pattern for Each Submicron Segment

• Local Structure, Orientation, Full Strain Tensor, Deformation Microstructure

*Measurements performed on Sector 34 ID-E, Advanced Photon Source, ANL

Measurements on Compressed Cu

CCD

Cu[001]

• Initially Dislocation- Free Copper Crystal

• Compression Deformed 1% and 2.3%

Microbeam

Profiler

Y[010]Parallelogram

shaped slices

Z[001]

Schematic of 3D X-ray Microscopy Deformation Microstructure Measurement

Measuring face normal to compressionaxis

Dislocation DynamicsSimulation of Deformationin [001] strained Cu

Direct linkage

10 µmBox

5 µmBox

Cu[001] R= (Rx,Ry,Rz)

X

Z

Y

Dislocation Dynamics Simulations of the Rotation Probability Distribution in 1.6% Strained Cu

Submicron (0.5 µm) Spatially Resolved Rotation Measurements for 2.3% Compression Strained [001] Cu

• Initial Quantitative Comparison of Rotation Probability Distribution Measurements withDislocation Dynamics Simulations for [001] Compression Strain in Copper

• Correlations longer than 10 μm in both 1% and 2.3% strained samples.

• Rotations ~2x larger in 2.3% strained sample

• Increased substructure refinement within 2.3% strained sample

Full range

The research at Florida State University and ORNL was supported by the DOE Office of Basic Energy Sciences Division of Materials Sciences and Engineering. The Advanced Photon Source is supported by the US Department of Energy Office of Science, Basic Energy Sciences.

• All stress components are distributed similarly.

• Stress fluctuations increase as the strain level and dislocation density increase.

• Distributions are dependent on the dislocation density and related to its correlation.

• Anisotropic distribution indicates stress patterning.

• The probability distributions for dislocation density tensor components in deformed Cu are symmetric with widths that increase linearly with strain.

• The dislocation density tensor is determined mainly by the curvature part in deformed metals.

Left: Dislocation density tensor component α12 (mrad/µm)

Right: Curvature contribution

• Statistical analysis shows the internal elastic fields and dislocation density tensor are distributed anisotropically, with zero mean value and strain-dependent fluctuations.

• The internal elastic fields exhibit long-range correlations that are related to the distribution and correlations within the underlying dislocation structure.

• The analysis shows that the dislocation density tensor components exhibit symmetric distributions, which can be attributed to the zero mean lattice curvature and the statistical homogeneity of deformation process.

• The analysis has also shown that local curvature is the main contributor to the dislocation density tensor in deformed metals.

• Initial comparison between 3D x-ray microscopy measurements and dislocation dynamics simulations of deformation in Cu show semi-quantitative agreement, but also indicate the need for larger simulation volumes and measurements with improved spatial resolution.

X

Single Plane Dist. (5 x 5 µm2)

+/- 0.06o

40 30 20 10

Z (µm)

-50-40

-30-20

-100

10Y (µm

)

-1.0

-0.5

0.0

0.5

1.0

x10-3

Cu (001)

2.3% S

trainM

eas. (0.5 µm

Res.)

Single Plane Rotation Plots

• Rotation probability distributions show non-symmetric distributions for measurements and simulations • Single plane 5 x 5 µm2 rotation probability distributions are more narrow than full sample measurements and simulations• Impact of 0.5 µm measurement spatial resolution on the rotation probability distribution is moderate

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